1. Applications of Frequency Extraction to Cavity Modeling Travis M. Austin* and John R. Cary*,¶ Tech-X Corporation*, University of Colorado-Boulder ¶ Collaborator: Leo Bellantoni, FNAL UCLA COMPASS Meeting 3 December 2008

2. Overview Tech-X Corporation 2 Background on Cavity Modeling Finite-Difference Time-Domain Simulations Frequency Extraction Algorithm Filtered Excitation Filter-Diagonalization Verification of Spherical Cavity Validation of A15 Cavity Conclusion

3. Cavity Modeling Verifying and validating EM codes is a crucial part of cavity modeling since it provides evidence of the effectiveness of the code COMPASS codes like Omega3p have made a concerted effort at V&V We focus in this talk on V&V efforts for Tech-X Corporation’s VORPAL code VORPAL has been successful in the past at laser wakefield simulations and electron cooling Tech-X Corporation 3

4. Cavity Modeling Machining is accurate to about 1 mil or 0.0254 mm Results in [Burt et al., 2007] showed frequencies to be sensitive to equatorial radius by about 80 MHz/mm for a deflecting cavity Machining can produce cavities with frequencies shifted by about ± 2 MHz from the original specs. Careful remeasurements after fabrication can be using simulations instead of bead pull experiments if the simulations are accurate. Tech-X Corporation 4

5. Finite-Difference Time-Domain Tech-X Corporation 5 ExEx ExEx EzEz EzEz EyEy EyEy EyEy BxBx ByBy BzBz EzEz...... ExEx Maxwell’s Equations “Rectangular Grid”

6. Finite-Difference Time-Domain Tech-X Corporation 6 Embedded Boundary Methods

7. Finite-Difference Time-Domain Curved domains described analytically These domains are not represented by the logically rectangular domain in contrast to unstructured FE meshes There are three methods for representing contribution of curved boundaries for logically rectangular domains: –Stairstep –Dey-Mittra –Zagorodnov Stairstep and Dey-Mittra discussed on next page Zagorodnov only recently implemented Tech-X Corporation 7 Embedded Boundary Methods

8. Finite-Difference Time-Domain Tech-X Corporation 8 (a) Stairstep Approach (b) Dey-Mittra Approach - Only change Faraday update

9. Finite-Difference Time-Domain FDTD is a second-order method Curved domains modeled using embedded boundary methods Embedded boundary method requires adjusting lengths (l ij, l ik, l jk ) and areas (a ij ) used in the Faraday update step Faces with small area excluded from computations to minimize the reduction in time-step due to CFL Method maintains second-order in time and space unless too many cells thrown out Tech-X Corporation 9

10. VORPAL Computational Framework Tech-X Corporation 10 Based on the FDTD method Mainly uses the Dey-Mittra method for embedded boundaries Excellent scaling on >10000 processors of Franklin for EM problem with ~200 million grid points Load balancing and ADI methods currently being investigated for even better performance in the future

11. Frequency Extraction Algorithm Tech-X Corporation 11 Eigenvalue problems typically consist of constructing a large matrix system and using an iterative method to find the eigenvalues Robust eigenvalue solver is necessary to compute the eigenvalues in a reasonable time These methods require more memory (storing matrix and multiple vectors) and are generally less scalable than FDTD methods Goal is to construct an eigenvalue solver (or frequency extraction algorithm) that depends on FDTD methods which are very scalable and require minimal memory

12. Frequency Extraction Algorithm Tech-X Corporation 12 Use FDTD method as it scales well for massively parallel machines like the NERSC machine Franklin Extract frequencies through Filter to desired modes Determine subspace with SVD Diagonalize in subspace Get multiple modes at once G.W Werner and J.R Cary, J. Comp. Phys., 227, 5200-5214, 10, (2008).

13. Frequency Extraction Algorithm Tech-X Corporation 13 Consider Then where Use that vanishes for t > T, where T is the excitation time, i.e., where For the range, we use

14. Frequency Extraction Algorithm Tech-X Corporation 14 Obtain L state vectors (s l ) for L > M, the number of modes, which correspond to evaluation of the field at L times for t > T and define r l = Hs l Determine the approximate number of modes, M, in the range Evaluate (s l ) at P random points on the grid to obtain P X L matrix S and the P X L matrix R such that R and S may be overdetermined so solve instead Find the SVD of Find the singular values of Frequencies are calculated as

15. Frequency Extraction Algorithm Degeneracies (or near degeneracies) can be extracted with multiple simulations to generate the state vectors (s l ) Once the state vectors are generated from FDTD simulations, the frequency extraction algorithm is quick (< 1min) Constructing the spatial mode patterns for each frequency also takes only several minutes depending on problem size Results in [Cary and Werner, 2008] verified method for 2D rectangular wave guide Tech-X Corporation 15

16. Validation of Sphere Tech-X Corporation 16 Simulation parameters 18 degree slice of a spherical cavity Radius = 0.1 m Grid size = 2 mm Frequency range = 2 ~ 4 GHz Expected modes (TEnmp)‏ TE101 2.14396 GHz TE201 2.74995 GHz TE301 3.33418 GHz TE102 3.68598 GHz TE401 3.90418 GHz

17. Validation of Sphere Tech-X Corporation 17 ModeAnalytical (GHz) Calculated (GHz) Rel. Error TE1012.143962.145500.00072 TE2012.749952.750910.00035 TE3013.334183.333780.00012 TE1023.685983.684580.00038 TE4013.904183.903020.00030 These preliminary results have similar accuracy to HFSS and Microwave Studio. Omega3p more accurate by a three orders of magnitude. (HFSS, Microwave Studio, and Omega3p results obtain from JLab. VORPAL results produced by Seah Zhou of Tech-X Corp.)

18. Validation of A15 Cavity Tech-X Corporation 18 Compute frequencies for 9-cell crab cavity and compare to MAFIA/MWS Crab cavity squashed in the z-direction to eliminate degeneracies Simulations with up to 25 million cells Extrapolated results consistently differ from MAFIA/MWS by ~3 MHz Background

19. Validation of A15 Cavity Tech-X Corporation 19 A15 Cavity is an aluminum cavity fabricated at Fermilab in 1999 Designed for development of a K+ beam It has been extensively tested, measured, and simulated Simulations performed by MAFIA considered computing frequencies of accelerating and deflecting modes Tech-X using VORPAL has concentrated on the deflecting (TM 110 ) modes from the A15

20. Validation of A15 Cavity Tech-X Corporation 20 Equator Radius: 47.19 mm Iris Radius: 15.00 mm Cavity Length: 153.6 mm Cavity contains end plate holes used for bead pull experiments and for creating dipoles Five Deflecting Modes: f0f0 f1f1 f2f2 f3f3 f4f4 3902.8103910.4043939.3364001.3424106.164

21. Validation of A15 Cavity Tech-X Corporation 21 Excitation Pattern:

22. Validation of A15 Cavity Tech-X Corporation 22 Simulation Parameters: Two simulations used to capture degeneracies Excitation time: 100 periods @ 4 GHz Total simulation time: 150 periods @ 4 GHz Max number of grid points: ~20 million grid points Max Total Time Steps: 437369 time steps

23. Validation of A15 Cavity Tech-X Corporation 23

24. Validation of A15 Cavity Tech-X Corporation 24 Relative Error of Deflecting Modes Computed by VORPAL: f0f0 f1f1 f2f2 f3f3 f4f4 6.5e-44.9e-45.1e-41.2e-36.1e-4 Relative Error of Deflecting Modes Computed by MAFIA: f0f0 f1f1 f2f2 f3f3 f4f4 1.4e-31.3e-3-------

25. Validation of A15 Cavity VORPAL was too low by 2 MHz for the p mode MAFIA was too low by 5 MHz for the p mode MAFIA calculations were too large on spacing between the p deflection mode and the next higher mode by 6.41% and VORPAL calculations were too large by 7.6% Possible causes for differences between calculations and experimental measurements: –Failed to account for atmospheric conditions –End plate holes lead to frequency shift –Discrepancies between specs and machining Tech-X Corporation 25

26. A15 Accelerator Cavity Computations Tech-X Corporation 26

27. Validation of A15 Cavity Tech-X Corporation 27 Relative Error of Deflecting Modes Computed by VORPAL for 0.03 mm smaller equatorial radius: f0f0 f1f1 f2f2 f3f3 f4f4 6.5e-44.9e-45.1e-41.2e-36.1e-4 Relative Error of Deflecting Modes Computed by VORPAL for original equatorial radius: f0f0 f1f1 f2f2 f3f3 f4f4 5.6e-51.4e-57.3e-57.0e-57.6e-5

28. A15 Cavity Computations Tech-X Corporation 28 3902.810 MHz (p mode)3910.404 MHz 4001.342 MHz3939.336 MHz

29. Complete Picture of p deflection mode Tech-X Corporation 29

30. Final Remarks Thanks to Leo Bellantoni at FNAL for assisting on verification study Working with Jlab on further validation for sphere and examining maximum value of B field on surface We are currently working on a paper which will be submitted soon showcasing this work Future topics consist of using algorithm in an optimization loop for cavity design. Tech-X Corporation 30